Results 1  10
of
43
Boltzmann equation without angular cutoff in the whole space: II, global existence for hard potential, to appear in Analysis and Applications
"... ar ..."
(Show Context)
Optimal largetime behavior of the VlasovMaxwellBoltzmann system in the whole space
"... In this paper we study the largetime behavior of classical solutions to the twospecies VlasovMaxwellBoltzmann system in the whole space R³. The existence of global in time nearby Maxwellian solutions is known from [37] in 2006. However the asymptotic behavior of these solutions has been a chal ..."
Abstract

Cited by 29 (20 self)
 Add to MetaCart
In this paper we study the largetime behavior of classical solutions to the twospecies VlasovMaxwellBoltzmann system in the whole space R³. The existence of global in time nearby Maxwellian solutions is known from [37] in 2006. However the asymptotic behavior of these solutions has been a challenging open problem. Buildingon ourprevious work[12]on timedecay for the simpler VlasovPoissonBoltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of O(t − 3 2 in L2 ξ (Lrx + 3
The VlasovPoissonLandau System in a Periodic Box
, 2011
"... The classical VlasovPoissonLandau system describes dynamics of a collisional plasma interacting with its own electrostatic …eld as well as its grazing collisions. Such grazing collisions are modeled by the famous Landau (FokkerPlanck) collision kernel, proposed by Landau in 1936. We construct glo ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
The classical VlasovPoissonLandau system describes dynamics of a collisional plasma interacting with its own electrostatic …eld as well as its grazing collisions. Such grazing collisions are modeled by the famous Landau (FokkerPlanck) collision kernel, proposed by Landau in 1936. We construct global unique solutions to such a system for initial data which have small weighted H 2 norms, but can have large H k (k 3) norms with high velocity moments. Our construction is based on accumulative study on the Landau kernel in the past decade [G1] [SG13], with four extra ingredients to overcome the speci…c mathematical di ¢ culties present in the VlasovPoissonLandau system: a new exponential weight of electric potential to cancel the growth of the velocity, a new velocity weight to capture the weak velocity di¤usion in the Landau kernel, a decay of the electric …eld to close the energy estimate, and a new bootstrap argument to control the propagation of the high moments and regularity with large amplitude. 1
Global existence and full regularity of the Boltzmann equation without angular cutoff
 Comm. Math. Phys
"... ar ..."
(Show Context)
Optimal time decay of the non cutoff Boltzmann equation in the whole space
, 2010
"... Abstract. In this paper we study the largetime behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cutoff assumption in the whole space Rn x with n ≥ 3. We use the existence theory of global in time nearby Maxwellian solutions from [13 ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we study the largetime behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cutoff assumption in the whole space Rn x with n ≥ 3. We use the existence theory of global in time nearby Maxwellian solutions from [13,14]. It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cutoff assumption [3, 29]. For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal largetime
Hypoelliptic estimates for a linear model of the Boltzmann equation without angular cutoff
"... Abstract. In this paper, we establish optimal hypoelliptic estimates for a class of kinetic equations, which are simplified linear models for the spatially inhomogeneous Boltzmann equation without angular cutoff. 1. ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper, we establish optimal hypoelliptic estimates for a class of kinetic equations, which are simplified linear models for the spatially inhomogeneous Boltzmann equation without angular cutoff. 1.
On measure solutions of the Boltzmann equation, part I: moment production and stability estimates
 J. Differential Equations
"... Abstract. ThespatiallyhomogeneousBoltzmannequationwith hardpotentialsisconsidered for measure valued initial data having finite mass and energy. We prove the existence of weak measure solutions, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approxim ..."
Abstract

Cited by 11 (5 self)
 Add to MetaCart
(Show Context)
Abstract. ThespatiallyhomogeneousBoltzmannequationwith hardpotentialsisconsidered for measure valued initial data having finite mass and energy. We prove the existence of weak measure solutions, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions. Mathematics Subject Classification (2000): 35QEquationsofmathematicalphysics and other areas of application [See also 35J05, 35J10, 35K05, 35L05], 76P05 Rarefied gas
SHARP ANISOTROPIC ESTIMATES FOR THE BOLTZMANN COLLISION OPERATOR AND ITS ENTROPY PRODUCTION
, 1007
"... Abstract. This article provides sharp constructive upper and lower bound estimates for the Boltzmann collision operator with the full range of physical non cutoff collision kernels (γ> −n and s ∈ (0,1)) in the trilinear L 2 (R n) energy 〈Q(g,f),f〉. These new estimates prove that, for a very gene ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
(Show Context)
Abstract. This article provides sharp constructive upper and lower bound estimates for the Boltzmann collision operator with the full range of physical non cutoff collision kernels (γ> −n and s ∈ (0,1)) in the trilinear L 2 (R n) energy 〈Q(g,f),f〉. These new estimates prove that, for a very general class of g(v), the global diffusive behavior (on f) in the energy space is that of the geometric fractional derivative seminorm identified in the linearized context in our earlier works [15,16]. We further prove new global entropy production estimates with the same anisotropic seminorm. This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non cutoff Boltzmann collision operator in the energy space L 2 (R n). Contents
The Boltzmann equation, Besov spaces, and optimal time decay rates in the whole space. Preprint 2012. See also arXiv:1206.0027
"... ar ..."
(Show Context)